When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2023.1
URN: urn:nbn:de:0030-drops-186545
URL: https://drops.dagstuhl.de/opus/volltexte/2023/18654/
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On Hashing by (Random) Equations (Invited Talk)

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Abstract

The talk will consider aspects of the following setup: Assume for each (key) x from a set š° (the universe) a vector a_x = (a_{x,0},ā¦ ,a_{x,{m-1}}) has been chosen. Given a list Z = (z_i)_{i ā [m]} of vectors in {0,1}^r we obtain a mapping Ļ_Z: š° ā {0,1}^r, x ā¦ āØa_x,Zā© := āØ_{i ā [m]} a_{x,i} ā z_i, where āØ is bitwise XOR. The simplest way for creating a data structure for calculating Ļ_Z is to store Z in an array Z[0..m-1] and answer a query for x by returning āØ a_x,Zā©. The length m of the array should be (1+Īµ)n for some small Īµ, and calculating this inner product should be fast. In the focus of the talk is the case where for all or for most of the sets S ā š° of a certain size n the vectors a_x, x ā S, are linearly independent. Choosing Z at random will lead to hash families of various degrees of independence. We will be mostly interested in the case where a_x, x ā š° are chosen independently at random from {0,1}^m, according to some distribution š. We wish to construct (static) retrieval data structures, which means that S ā š° and some mapping f: S ā {0,1}^r are given, and the task is to find Z such that the restriction of Ļ_Z to S is f. For creating such a data structure it is necessary to solve the linear system (a_x)_{x ā S} ā Z = (f(x))_{x ā S} for Z. Two problems are central: Under what conditions on m and š can we expect the vectors a_x, x ā S to be linearly independent, and how can we arrange things so that in this case the system can be solved fast, in particular in time close to linear (in n, assuming a reasonable machine model)? Solutions to these problems, some classical and others that have emerged only in recent years, will be discussed.

BibTeX - Entry

```@InProceedings{dietzfelbinger:LIPIcs.ESA.2023.1,
author =	{Dietzfelbinger, Martin},
title =	{{On Hashing by (Random) Equations}},
booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
pages =	{1:1--1:1},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-295-2},
ISSN =	{1868-8969},
year =	{2023},
volume =	{274},
editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},